Solvable points on genus one curves
Whether a polynomial in one variable with rational coefficients has a root in a solvable extension of the rationals is more or less known, after the work of Galois. For polynomials in two variables (or more), one can ask a similar question: Let \(X\) be a smooth geometrically irreducible projective curve of genus \(g\) defined over a field \(F\); is it true that \(X\) has a point defined over some solvable extension of \(F\)? For some local fields \(F\), \textit{A. Pál} [Can. J. Math. 56, No. 3, 612--637 (2004; Zbl 1066.14032)] has given both positive and negative results, depending on the genus \(g\) of the curve. In the present paper the authors consider curves of genus one defined over the rational numbers. Let \(C\) be such a curve, \(E\) be its Jacobian and \(L(E,s)\) be the \(L\)-series associated to \(E\). Assume that \(C({\mathbb Q}_p)\neq \emptyset\) for all primes \(p\). The following theorems are proved: Theorem 1. If \(L(E,s)\) has a zero of order 0 or 1 at \(s=1\), then \(C\) has a point over a solvable extension of \({\mathbb Q}\). Theorem 2. If \(E = \text{Jac}\;C\) is semistable, then \(C\) has a point over a solvable extension of \({\mathbb Q}\). The proofs rely on quite a number of very deep ideas; among them, the use of the Kolyvagin constructions of the ramified classes in \(H^1(\overline{\mathbb Q}/{\mathbb Q},E_{p^n})\) together with the general principle, explained in the introduction of [\textit{A. Wiles}, Ann. Math. (2) 141, 443--551 (1995; Zbl 0864.11029)], that if one can construct enough ramified classes, then the unramified classes are already contained in the group generated by those ramified classes.
scientific article; zbMATH DE number 5278249
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